A Class of $(n, k, r, t)$ IS-LRCs Via Parity Check Matrix

A code is called $(n, k, r, t)$ information symbol locally repairable code (IS-LRC) if each information coordinate can be achieved by at least $t$ disjoint repair sets containing at most $r$ other coordinates. This letter considers a class of $(n, k, r, t)$ IS-LRCs, where each repair set contains exactly one parity coordinate. We explore the systematic code in terms of the standard parity check matrix. First, we propose some structural features of the parity check matrix by showing a connection with the membership matrix. After that, we place parity check matrix based proof of several bounds associated with the code. In addition, we provide two constructions of optimal parameters of $(n,k,r,t)$ IS-LRCs with the help of two Cayley tables of a finite field. Finally, we present a generalized result on optimal $q$-ary $(n,k,r,t)$ IS-LRCs related to MDS codes.